7701 (Number Theory)
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2015–2016
MATH7701
Intermediate
Half unit (= 7.5 ECTS credits)
2
3 hour lectures per week
90% examination, 10% coursework
MATH1202, (MATH2201 recommended),
Dr RM Hill
Course Description and Objectives
This course is an introduction to elementary number theory. The main focus is on solving
equations and congruences in integers, although various other rings will appear in the proofs of
theorems.
Recommended Texts
No book does exactly what is in the course, but the following books might be useful:
(i) D. M. Burton, “Elementary Number Thery”;
(ii) H. E. Rose, “A course in number theory”;
(iii) K. Ireland and M. Rosen, “A classical introduction to modern number theory”;
(iv) H. Davenport, “The higher arithmetic” (this is mainly for the earlier parts of the course,
and also for the final section on continued fractions);
(v) J.–P. Serre, “A course in arithmetic” (this is a bit more advanced than is needed for the
course).
Detailed Syllabus
The Euler totient function ϕ. We’ll show how to calculate ϕ(n). Using this, we’ll be able
to calculate powers of integers modulo n, and solve congruences of the form xa ≡ b mod n.
Existence of primitive roots. In this section we’ll prove that for any prime number p, the
multiplicative group F×
p is cyclic.
Quadratic reciprocity. Given an integer a, we’ll answer the question: for which primes p is
a a square modulo p?
Hensel’s Lemma. Suppose f is a polynomial, and we have a solution to f (x) ≡ 0 mod p.
We’ll show how we can modify the solution to get a solution to f (x) ≡ 0 mod pn .
Power series modulo powers of primes. We’ll introduce the idea of a power series modulo
pn . In particular introduce the logarithm and exponential functions on Z/pn and prove their
properties. We also decompose the group (Z/pn )× as a direct sum of the exponentials and the
Teichmüller representatives. Introduction to the p-adic integers.
Factorization in quadratic rings. The Gaussian integers are numbers of the form x + iy
where x and y are integers. These numbers form a ring, and this is an example of a quadratic
ring. We’ll study quadratic rings in general, and prove that in a number of cases they have
unique factorization. We’ll also show how to factorize a prime number in a quadratic ring, using
the reciprocity law.
Continued fractions. We’ll describe the continued fraction expansion of a real number. As a
consequence we’ll find a method for solving Pell’s equation x2 − dy 2 = 1. This allows us to find
all the units in a real quadratic ring.
May 2015 MATH7701